Sign variation, the Grassmannian, and total positivity

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• 作者：Steven N. Karp¹ ; ^{skarp@berkeley.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Sign variation ; Total positivity ; Totally nonnegative Grassmannian ; Amplituhedron ; Grassmann polytope ; Positroid ; Oriented matroid
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：145
• 期：Complete
• 页码：308-339
• 全文大小：631 K

文摘

The totally nonnegative Grassmannian is the set of k-dimensional subspaces V   of Rⁿ${\mathbb{R}}^n$ whose nonzero Plücker coordinates all have the same sign. Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that V is totally nonnegative iff every vector in V, when viewed as a sequence of n   numbers and ignoring any zeros, changes sign at most k−1$k-1$ times. We generalize this result from the totally nonnegative Grassmannian to the entire Grassmannian, showing that if V is generic (i.e. has no zero Plücker coordinates), then the vectors in V change sign at most m times iff certain sequences of Plücker coordinates of V   change sign at most m−k+1$m-k+1$ times. We also give an algorithm which, given a non-generic V whose vectors change sign at most m times, perturbs V into a generic subspace whose vectors also change sign at most m times. We deduce that among all V whose vectors change sign at most m times, the generic subspaces are dense. These results generalize to oriented matroids. As an application of our results, we characterize when a generalized amplituhedron construction, in the sense of Arkani-Hamed and Trnka (2013), is well defined. We also give two ways of obtaining the positroid cell of each V in the totally nonnegative Grassmannian from the sign patterns of vectors in V.